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Can 1/3 and 1/3 = 2/6? It seemed so

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Re: Can 1/3 and 1/3 = 2/6? It seemed so

#281

Post by wcarey »

08-15 wrote: post #181

That's what happens when you try to "build an intuition" for a concept instead of defining it. Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? That don't make no sense!! The whole point of common fractions is to close integer…

As someone who has pushed pretty hard towards precise definition and rigor as a teacher, you're in the (happy!) minority being able to approach mathematics that way. For lots of students, the process of moving from definitions and simple rules to other statements is extraordinarily difficult.

I teach the properties of exponents to 14 year olds as a unit on logical necessity - the properties all flow necessarily from the definition of the exponent. About a third of the students say, "cool" and never miss anything on any assessment again because the answers flow necessarily from the givens.

About a third work their way through it fine.

About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 despite working many particular examples where that is manifestly false.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#283

Post by trhway »

Wowfunhappy wrote: post #111
forbiddenvoid wrote: post #40

The units are missing, and I think that's a key factor here. Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions. The student came up and wrote 1/3 + 1/3 = 2/6. What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those t…

You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard! > The confusion comes because no one calls out that they're talking about fractions of different things. But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking abo…

Just a missing step in the explanation:

"1/3 of the students at the first table + 1/3 of the students at a second table =

1/3 of 1/2 of all the students at both tables [ie. those students at the first table] + 1/3 of another 1/2 of all the students at both tables [ie. those at a second table] =

1/3 of the students at both tables"

In US math education I noticed there is a lot emphasis on specific patterns/formulas/etc., kind of "deus ex machina", and a lot of excercises to mechanize their application, yet too little is spend on where it comes from, manual derivation/proval of formulas, establishing logical connections, etc.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#284

Post by EmilioMartinez »

My take: Adding proportions is kinda "semantically incorrect". It's well defined as a value, it just doesn't mean what you think it means, akin to thinking that union of two sets is written A + B (which would be the set of pair-wise additions of elements). Now, dovetailing the equation with units that signify concrete quantities (as suggested by others) circumvents this by making it meaningful, at the cost of no longer working with the abstract concept of proportion.

An example of semantically meaningful manipulation of proportions is the weighted combination of proportions. This here's a more verbose version of what the teacher wrote: (3/6)* (1/3) + (3/6)*(1/3) = 1/3 (that is, the weighted average of the proportions give the final proportion)

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#285

Post by lucio »

When you put two tables together, each table has 1/2 of the total number of kids. By having 2 tables, now you have 6 kids, so:

the original 1/3 in the first table becomes 1/6

the original 1/3 in the second table becomes 1/6

so adding the two tables means 2/6 are girls

She needs to explain 1/3 of 1/2 (fraction of a fraction) before using an example where she joins two units (two tables)

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#286

Post by djmips »

As the concept of fairness is innate in primates including humans in the 4th grade I wouldn't stop and correct the students but try and re-frame the problem in terms of a problem where fairness was involved and then immediately the ratios and the units would come into more clear focus. As an example I might say that if I promised to give you 1/3 of the pizza as a reward for cleaning your room. And then how much should you get if I had 1 pizza. How about 2 pizzas? Things like that.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#287

Post by RHSeeger »

PaulDavisThe1st wrote: post #256
RHSeeger wrote: post #246

Earlier quoted context omitted.

I have a hard time accepting this. My daughter is in 3rd grade and seems to have a reasonable grasp of fractions. They teach her about them at (public) school and we've discussed them at home. Sure, "1/3 + 2/7" is outside her reach, but simpler stuff, things she can visualize, are well within her grasp. > why is 2/3 different from 2/6 anyway Because 2 pieces of a pie that you cut into 3 pieces is more than 2 pieces o…

No disrespect, but you read HN and think that your daughter's take on math and the home context you provide for it is a sensibly representative starting point? My daughter's now 25 and has been quite the nerd herself through the years (eventually landing in linguistics and speech pathology), but I'd never assume that the fascinations she had instrinsically and that I helped foster as a parent were really typical. I w…

I think the homework her school gives out is a sensibly representative starting point. Why wouldn't it be?

That homework expects a reasonable understanding of fractions. Enough that the child doing them can understand the difference between the number of slices in a pie being representative of the bottom number of a fraction.

Sure, I do math exercises with my daughter that aren't representative of what they teach in school (square roots, the fact that parallel lines _do_ meet at the vanishing point in the real world, etc). But those things aren't what I'm basing my assumptions are; the expectations the school has of her are.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#288

Post by C4stor »

wcarey wrote: post #14
mkl wrote: post #7

Earlier quoted context omitted.

I think x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.

Right. Of course, if you write x(3), other mathematicians should frown at you because you're making bad notational choices. It's a bit like explaining to students that the real way to know which 3rd declension nouns are i-stems in Latin is to say the genitive plural both ways. The one that doesn't sound wrong is correct. But you have to have a lot of time in the language for that to work.

Also, there are actual quite reasonable rules to know which 3rd declension nouns are i-stems, so it doesn't seem too right too just say "follow your gut".

Btw, I wouldn't abuse latin comparisons on an american forum, I don't think it's quite in the culture ^^

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#289

Post by hoppla »

TeMPOraL wrote: post #257
Wowfunhappy wrote: post #111

Earlier quoted context omitted.

You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard! > The confusion comes because no one calls out that they're talking about fractions of different things. But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking abo…

I think we really need to start teaching people paying attention to units. Even in math classes. 1/3 + 1/3 is not a correct mathematical description of the problem. 1/3 [students at table A] + 1/3 [students at table B] is. Let's shorten this to: 1/3 sTA + 1/3 sTB. Then, you factor out the 1/3, to get: 1/3 * (sTA + sTB). And now it's it's impossible to give the wrong answer "2/3". Or, in other words, it's best to "kee…

Units is a real eye opener once you do it right. I remember when i studied and could almost verify what I did was correct, I would just check if the resulting unit was what I expected

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#290

Post by [deleted] »

TeMPOraL wrote: post #257
Wowfunhappy wrote: post #111

Earlier quoted context omitted.

You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard! > The confusion comes because no one calls out that they're talking about fractions of different things. But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking abo…

I think we really need to start teaching people paying attention to units. Even in math classes. 1/3 + 1/3 is not a correct mathematical description of the problem. 1/3 [students at table A] + 1/3 [students at table B] is. Let's shorten this to: 1/3 sTA + 1/3 sTB. Then, you factor out the 1/3, to get: 1/3 * (sTA + sTB). And now it's it's impossible to give the wrong answer "2/3". Or, in other words, it's best to "kee…

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